Zero-sum Games
Michael Bacharach · Game Theory · 1989
Zero-sum games are to the theory of games what the twelve-bar blues is to jazz: a polar case, and a historical point of departure. A game is a situation in which (i) each of a number of agents ( players ) has a set of alternative courses of action ( strategies ) at his disposal; (ii) there are outcomes which depend on the combination of the players’ actions and give rise to preferences by the players over these combinations; (iii) the players know, and know that each other knows, these preferences. (Strictly, such a situation is a game of complete information in normal form : these qualifications should hencefore be understood.) In the case which dominates the literature of zero-sum games there are two players, A and B say, each with a finite set of strategies, and their preferences can be represented by von Neumann-Morgenstern utilities. The preference structure can then be displayed in a payoff matrix whose ( i , j )th entry ( u ij , v ij ) gives the expected utilities or payoffs of A and B respectively for A using his ith strategy and B using his jth. A game of this type in which u ij + v ij = 0 for all i,j is known as a zero-sum matrix game (henceforth simply zero-sum game ). In a zero-sum game the players have exactly opposed preferences over strategy-pairs. Hence there is no scope for the pair of them to act as a pair — there is nothing for them to cooperate about. The theory of cooperative zero-sum games is thus an empty box; zero-sum games are noncooperative games, and each player must choose in uncertainty of the other’s choice. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.